Radial basis function approximations as smoothing splines

Abstract

Radial basis function methods for interpolation can be interpreted as roughness-minimizing splines. Although this relationship has already been established for radial basis functions of the form g( r)= r α and g(r)=r α log (r) , it is extended here to include a much larger class of functions. This class includes the multiquadric g( r)=( r 2+ c 2) 1/2 and inverse multiquadric g( r)=( r 2+ c 2) −1/2 functions as well as the Gaussian exp (−r 2/D) . The crucial condition is that the Fourier transform of g(| x|) be positive, except possibly at the origin. The appropriate measure of roughness is defined in terms of this Fourier transform. To allow for possibility of noisy data, the analysis is presented within the general framework of smoothing splines, of which interpolation is a special case. Two diagnostic quantities, the cross-validation function and the sensitivity, indicate the accuracy of the approximation.